Mathematical Operations — Symbol Substitution, BODMAS Tricks, and Coded Inequalities

Mathematical operations questions test your ability to decode symbols, apply BODMAS correctly, and interpret coded inequalities. This topic appears in almost every Kerala PSC exam (1-3 questions). Master the patterns below.

1. Symbol Substitution — Concept

In these questions, mathematical symbols (+, -, x, /) are replaced with other symbols or words. You must substitute the correct operations and solve.

Golden Rule: Always apply BODMAS after substitution.

BODMAS Order	Operation
B	Brackets (solve first)
O	Orders (powers, roots)
D	Division
M	Multiplication
A	Addition
S	Subtraction

2. Solved Problems — Symbol Substitution (Problems 1-5)

Problem 1

If + means x, - means /, x means +, and / means -, find the value of: 8 + 6 - 3 x 12 / 4

Step	Working
Substitute symbols	8 x 6 / 3 + 12 - 4
Apply BODMAS: Division first	8 x 2 + 12 - 4
Multiplication next	16 + 12 - 4
Addition and Subtraction (left to right)	28 - 4 = 24

Answer: 24

Problem 2

If P means +, Q means -, R means x, and S means /, find: 18 R 6 S 3 P 5 Q 7

Step	Working
Substitute	18 x 6 / 3 + 5 - 7
Division first	18 x 2 + 5 - 7
Multiplication	36 + 5 - 7
Left to right	41 - 7 = 34

Answer: 34

Problem 3

If x stands for +, / stands for -, - stands for x, and + stands for /, find: 20 x 8 / 8 - 4 + 2

Step	Working
Substitute	20 + 8 - 8 x 4 / 2
Division first	20 + 8 - 8 x 2
Multiplication	20 + 8 - 16
Left to right	28 - 16 = 12

Answer: 12

Problem 4

If @ means +, # means -, $ means x, and % means /, find: 15 $ 4 % 2 @ 10 # 3

Step	Working
Substitute	15 x 4 / 2 + 10 - 3
Division first	15 x 2 + 10 - 3
Multiplication	30 + 10 - 3
Left to right	40 - 3 = 37

Answer: 37

Problem 5

If + means /, - means x, x means -, and / means +, what is the value of: 36 + 6 - 3 x 5 / 2?

Step	Working
Substitute	36 / 6 x 3 - 5 + 2
Division first	6 x 3 - 5 + 2
Multiplication	18 - 5 + 2
Left to right	13 + 2 = 15

Answer: 15

3. Solved Problems — BODMAS Trick Questions (Problems 6-10)

Problem 6

Which of the following interchanges will make the equation correct? 6 + 2 x 4 - 1 = 9

Test the options systematically:

Option Tested	Interchange	Result
Interchange + and x	6 x 2 + 4 - 1 = 12 + 4 - 1 = 15 (No)
Interchange + and -	6 - 2 x 4 + 1 = 6 - 8 + 1 = -1 (No)
Interchange x and -	6 + 2 - 4 x 1 = 6 + 2 - 4 = 4 (No)
Interchange + and -, also 6 and 1	1 - 2 x 4 + 6 = 1 - 8 + 6 = -1 (No)
Interchange x and -, also 4 and 1	6 + 2 - 1 x 4 = 6 + 2 - 4 = 4 (No)
Interchange 2 and 4, also + and -	6 - 4 x 2 + 1 = 6 - 8 + 1 = -1 (No)

Let us re-read: interchange + and - : 6 - 2 x 4 + 1 = 6 - 8 + 1 = -1. Try interchange x and - : 6 + 2 - 4 x 1 = 6 + 2 - 4 = 4. Try interchange numbers 2 and 4: 6 + 4 x 2 - 1 = 6 + 8 - 1 = 13. Try interchange numbers 1 and 2: 6 + 1 x 4 - 2 = 6 + 4 - 2 = 8. Try interchange numbers 4 and 1 and signs + and -: 6 - 2 x 1 + 4 = 6 - 2 + 4 = 8. Try interchange + and x: 6 x 2 + 4 - 1 = 12 + 4 - 1 = 15.

Corrected approach — interchange signs + and x AND numbers 6 and 4: 4 x 2 + 6 - 1 = 8 + 6 - 1 = 13. Still no. The answer depends on the specific options given in the exam. Key Technique: Systematically test each option using BODMAS.

Problem 7

If 5 * 3 = 16, 7 * 4 = 33, then 9 * 5 = ?

Pattern	Working
Check: a * b = a-squared - b-squared?	25 - 9 = 16 (Yes!), 49 - 16 = 33 (Yes!)
Apply to 9 * 5	81 - 25 = 56

Answer: 56 (Pattern: a * b = a-squared - b-squared)

Problem 8

If 2 # 3 = 13, 3 # 4 = 25, then 5 # 6 = ?

Pattern	Working
Check: a-squared + b-squared?	4 + 9 = 13 (Yes!), 9 + 16 = 25 (Yes!)
Apply to 5 # 6	25 + 36 = 61

Answer: 61 (Pattern: a # b = a-squared + b-squared)

Problem 9

Solve: 48 / 12 + 5 x 4 - 6 = ?

Step	Working
Division first	4 + 5 x 4 - 6
Multiplication	4 + 20 - 6
Left to right	24 - 6 = 18

Answer: 18

Problem 10

If 3 @ 2 = 7, 5 @ 3 = 13, 7 @ 4 = 19, then 9 @ 5 = ?

Pattern	Working
Check: a x 2 + b?	6 + 2 = 8 (No)
Check: a + b + (a - b)?	3 + 2 + 1 = 6 (No)
Check: a x b - (a - b)?	6 - 1 = 5 (No)
Check: 2a + b - 1?	6 + 2 - 1 = 7 (Yes!), 10 + 3 - 1 = 12 (No)
Check: a x b + 1?	6 + 1 = 7 (Yes!), 15 + 1 = 16 (No)
Check: a-squared - b?	9 - 2 = 7 (Yes!), 25 - 3 = 22 (No)
Check: a-squared - a + b?	9 - 3 + 2 = 8 (No)
Check: 2(a + b) - (a - b)?	2(5) - 1 = 9 (No)
Check: 2a + (a - b)?	6 + 1 = 7 (Yes!), 10 + 2 = 12 (No)
Check: (a + b) + (a - b) + 1?	5 + 1 + 1 = 7 (Yes!), 8 + 2 + 1 = 11 (No)
Check: a-squared - b-squared + b?	9 - 4 + 2 = 7 (Yes!), 25 - 9 + 3 = 19 (No—wait, 25 - 9 = 16, 16 + 3 = 19—No wait that IS 19 but let me check 5@3 first: 25-9+3=19, but answer should be 13)
Check: (a+b) x 2 - (b-1)?	10 - 1 = 9 (No)
Correct pattern: a x (b+1) - 2b?	3x3 - 4 = 5 (No)
Check: 3a - b?	9-2=7 (Yes!), 15-3=12 (No)
Check: 2a + 1?	7 (Yes!), 11 (No)
Pattern: a(a-1) + (b-1)?	3x2 + 1 = 7 (Yes!), 5x4 + 3-1? No: 20+2=22 (No)
Final: a x 2 + (a-b) x 1?	Review: 7, 13, 19 — differences are 6, 6. Arithmetic progression! 19 + 6 = 25

Answer: 25 (Pattern: results form an AP with common difference 6)

4. Solved Problems — Coded Inequalities (Problems 11-15)

In coded inequality questions, symbols represent relationships. You must decode and check which conclusions follow.

Common Coding Patterns

Symbol	Means
@	Greater than
#	Less than
$	Equal to
%	Greater than or equal to
and	Less than or equal to

Problem 11

Statements: A @ B, B $ C, C # D. Conclusions: (I) A @ C (II) D @ B

Step	Decode
A @ B	A is greater than B
B $ C	B equals C
C # D	C is less than D
Chain	A is greater than B = C is less than D, i.e., A is greater than B = C and C is less than D
Conclusion I: A @ C (A is greater than C)	A is greater than B and B = C, so A is greater than C. TRUE
Conclusion II: D @ B (D is greater than B)	C is less than D and B = C, so B is less than D, i.e., D is greater than B. TRUE

Answer: Both I and II follow

Problem 12

Statements: P % Q, Q # R, R $ S. Conclusions: (I) P @ S (II) S @ Q

Step	Decode
P % Q	P is greater than or equal to Q
Q # R	Q is less than R
R $ S	R equals S
Chain	P is greater than or equal to Q, Q is less than R = S
Conclusion I: P @ S (P is greater than S)	P is greater than or equal to Q, and Q is less than S. We cannot determine P vs S definitively. DOES NOT FOLLOW
Conclusion II: S @ Q (S is greater than Q)	S = R and R is greater than Q (since Q is less than R). So S is greater than Q. TRUE

Answer: Only II follows

Problem 13

If “A + B” means A is the father of B, “A - B” means A is the wife of B, “A x B” means A is the brother of B. What does “P + Q - R” mean?

Step	Decode
P + Q	P is the father of Q
Q - R	Q is the wife of R
Combined	P is the father of Q, and Q is the wife of R. So P is the father-in-law of R

Answer: P is the father-in-law of R

Problem 14

Statements: 5 @ 3, 3 % 7, 7 $ 2. If @ means is less than, % means is greater than, $ means is equal to. Conclusions: (I) 5 @ 7 (II) 3 @ 2**

Step	Decode
5 @ 3	5 is less than 3 (Take as given, even if mathematically odd — these are coded variables)
3 % 7	3 is greater than 7
7 $ 2	7 equals 2
Chain	5 is less than 3, 3 is greater than 7 = 2
Conclusion I: 5 @ 7 (5 is less than 7)	5 is less than 3, and 3 is greater than 7. So 5 is less than 3 and 7 is less than 3. Relationship between 5 and 7 is not certain. DOES NOT FOLLOW
Conclusion II: 3 @ 2 (3 is less than 2)	3 is greater than 7 and 7 = 2, so 3 is greater than 2. Therefore 3 is less than 2 is FALSE. DOES NOT FOLLOW

Answer: Neither I nor II follows

Problem 15

Statements: M $ N, N @ O, O % P. Conclusions: (I) M @ P (II) M @ O**

Step	Decode (using standard: @ = greater than, % = greater than or equal to, $ = equal to)
M $ N	M = N
N @ O	N is greater than O
O % P	O is greater than or equal to P
Chain	M = N is greater than O is greater than or equal to P
Conclusion I: M @ P (M is greater than P)	M = N is greater than O is greater than or equal to P. So M is greater than O and O is greater than or equal to P. Therefore M is greater than P. TRUE
Conclusion II: M @ O (M is greater than O)	M = N and N is greater than O. So M is greater than O. TRUE

Answer: Both I and II follow

5. Quick Tips for PSC Exams

Tip	Detail
Always write down substitutions first	Replace all symbols before calculating
BODMAS is non-negotiable	Even after symbol substitution, follow BODMAS strictly
In coded inequalities, build the chain	Link all statements into one chain, then test conclusions
Eliminate impossible conclusions	If the chain breaks (no clear link), the conclusion does not follow
Watch for “either/or” conclusions	If neither conclusion follows individually but one of them MUST be true logically, “either I or II” follows
Practice with a timer	These questions should take 30-45 seconds each in the exam

6. Practice Pattern Recognition

Given Pattern	Rule	Example
a * b = a-squared + b-squared	Sum of squares	3 * 4 = 9 + 16 = 25
a * b = a-squared - b-squared	Difference of squares	5 * 3 = 25 - 9 = 16
a * b = ab + a + b	Product plus sum	3 * 4 = 12 + 3 + 4 = 19
a * b = ab - (a + b)	Product minus sum	5 * 3 = 15 - 8 = 7
a * b = (a + b)(a - b)	Same as a-squared - b-squared	6 * 2 = 8 x 4 = 32
a * b = a-squared + ab	a(a+b)	3 * 2 = 9 + 6 = 15